The Equation of a Circle

A circle is the simplest of the conic sections: every point on it sits exactly the same distance from a single fixed centre. That one sentence is enough to derive its entire equation — and the equation, once you can read it, tells you the centre and radius at a glance.

Concept The definition and the three key lengths

Centre — the fixed point every point on the circle is measured from.

Radius

— that fixed distance.

Diameter — the longest chord, passing through the centre; always

Concept The standard form

Here is the centre and the radius. This is nothing more than the distance formula rearranged: the distance from to equals , squared on both sides to clear the square root.

Centred at the origin, this collapses to the simplest possible form:

With radius 1 as well, it becomes the unit circle,

— the foundation of trigonometry.

Watch the signs carefully. means , but is really , so . The right-hand side is , not — a circle written as has radius 5, not 25.

Example Writing the equation from centre and radius

Find the equation of the circle with centre and radius 6.

Substitute

into the standard form.

⟹ (x + 1)² + (y − 4)² = 36

The negative turned into a plus inside the bracket — the single most common place to slip up.

Example Reading centre and radius from an equation

Find the centre, radius and diameter of .

gives

;

, so

Diameter

⟹ centre (2, −3), radius 5, diameter 10

Concept The general form, and completing the square

Expanding the standard form always produces an equation of this shape. Going the other way — from general back to standard — requires completing the square separately on the terms and the terms. Only then can you read off the centre and radius.

Example Converting general form to standard form

Find the centre and radius of .

Group the variables and move the constant across:

Halve the

coefficient and square it: half of −6 is −3, and

Halve the

coefficient and square it: half of 8 is 4, and

Add both to each side:

Factor each bracket:

⟹ centre (3, −4), radius 6

Whatever you add to complete a square on the left must be added to the right as well — forgetting this is what produces a wrong radius.

Note Properties worth remembering

Circumference

and area

A central angle is twice the inscribed angle standing on the same arc.

An angle inscribed in a semicircle is always a right angle.

Chords of equal length sit equally far from the centre.

A tangent meets the circle at exactly one point, and is perpendicular to the radius drawn to that point.

Note How a line can meet a circle

Two points — the line is a secant.

One point — the line is a tangent.

No points — the line misses the circle entirely.

Two circles behave similarly: they may miss each other, touch at one point (externally or internally), or cross at two points — decided entirely by the distance between their centres compared with their radii.

Note Mistakes to avoid

Reading the right-hand side as

rather than

Getting the centre's signs backwards:

means

Adding the completing-the-square terms to only one side of the equation.

Confusing radius with diameter when a problem gives you one and asks for the other.

Not checking that

— if it comes out negative, no real circle exists.

Summary

A circle is the set of points a fixed distance r from a centre (h, k).
Standard form (x − h)² + (y − k)² = r² shows the centre and radius directly; the right-hand side is r², not r.
General form x² + y² + Dx + Ey + F = 0 hides them — complete the square on x and y to recover the standard form.
Centred at the origin the equation is x² + y² = r², and with r = 1 it is the unit circle used throughout trigonometry.